| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 13.1-a1 |
13.1-a |
$2$ |
$11$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
13.1 |
\( 13 \) |
\( 2^{12} \cdot 13^{11} \) |
$1.91217$ |
$(13,a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$11$ |
11B.3.1 |
$1$ |
\( 1 \) |
$4.479515803$ |
$0.975186933$ |
1.550518946 |
\( \frac{49503769856822467}{1792160394037} a + \frac{4663222451820445}{1792160394037} \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( -30 a + 469\) , \( -800 a - 2106\bigr] \) |
${y}^2+a{y}={x}^3-a{x}^2+\left(-30a+469\right){x}-800a-2106$ |
| 13.1-a2 |
13.1-a |
$2$ |
$11$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
13.1 |
\( 13 \) |
\( 2^{12} \cdot 13 \) |
$1.91217$ |
$(13,a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$11$ |
11B.3.1 |
$1$ |
\( 1 \) |
$0.407228709$ |
$10.72705626$ |
1.550518946 |
\( \frac{571}{13} a + \frac{45349}{13} \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( -11\) , \( a + 6\bigr] \) |
${y}^2+a{y}={x}^3-a{x}^2-11{x}+a+6$ |
| 13.2-a1 |
13.2-a |
$2$ |
$11$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
13.2 |
\( 13 \) |
\( 2^{12} \cdot 13 \) |
$1.91217$ |
$(13,a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$11$ |
11B.3.1 |
$1$ |
\( 1 \) |
$0.407228709$ |
$10.72705626$ |
1.550518946 |
\( -\frac{571}{13} a + \frac{45920}{13} \) |
\( \bigl[0\) , \( a - 1\) , \( a + 1\) , \( -11\) , \( -2 a + 7\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2-11{x}-2a+7$ |
| 13.2-a2 |
13.2-a |
$2$ |
$11$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
13.2 |
\( 13 \) |
\( 2^{12} \cdot 13^{11} \) |
$1.91217$ |
$(13,a+8)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$11$ |
11B.3.1 |
$1$ |
\( 1 \) |
$4.479515803$ |
$0.975186933$ |
1.550518946 |
\( -\frac{49503769856822467}{1792160394037} a + \frac{54166992308642912}{1792160394037} \) |
\( \bigl[0\) , \( a - 1\) , \( a + 1\) , \( 30 a + 439\) , \( 799 a - 2906\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(30a+439\right){x}+799a-2906$ |
| 26.1-a1 |
26.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
26.1 |
\( 2 \cdot 13 \) |
\( 2^{13} \cdot 13^{2} \) |
$2.27396$ |
$(2,a), (13,a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.600360525$ |
0.763158589 |
\( \frac{2737}{338} a - \frac{20367}{338} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -7 a - 17\) , \( a + 59\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a+1\right){x}^2+\left(-7a-17\right){x}+a+59$ |
| 26.1-a2 |
26.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
26.1 |
\( 2 \cdot 13 \) |
\( 2^{2} \cdot 13 \) |
$2.27396$ |
$(2,a), (13,a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.600360525$ |
0.763158589 |
\( -\frac{150489}{52} a + \frac{687965}{52} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -10\) , \( -a\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2-10{x}-a$ |
| 26.4-a1 |
26.4-a |
$2$ |
$2$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
26.4 |
\( 2 \cdot 13 \) |
\( 2^{13} \cdot 13^{2} \) |
$2.27396$ |
$(2,a+1), (13,a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.600360525$ |
0.763158589 |
\( -\frac{2737}{338} a - \frac{8815}{169} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( -6 a + 5\) , \( 5 a + 22\bigr] \) |
${y}^2+a{x}{y}={x}^3+a{x}^2+\left(-6a+5\right){x}+5a+22$ |
| 26.4-a2 |
26.4-a |
$2$ |
$2$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
26.4 |
\( 2 \cdot 13 \) |
\( 2^{2} \cdot 13 \) |
$2.27396$ |
$(2,a+1), (13,a+8)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.600360525$ |
0.763158589 |
\( \frac{150489}{52} a + \frac{134369}{13} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( -10\) , \( a - 1\bigr] \) |
${y}^2+{x}{y}={x}^3-a{x}^2-10{x}+a-1$ |
| 32.2-a1 |
32.2-a |
$2$ |
$17$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{42} \) |
$2.39512$ |
$(2,a), (2,a+1)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$17$ |
17B |
$1$ |
\( 2^{2} \) |
$0.514595823$ |
$2.451929956$ |
3.582798921 |
\( \frac{6357609}{131072} a + \frac{54062775}{131072} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( a + 27\) , \( a + 61\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a-1\right){x}^2+\left(a+27\right){x}+a+61$ |
| 32.2-a2 |
32.2-a |
$2$ |
$17$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{30} \cdot 11^{12} \) |
$2.39512$ |
$(2,a), (2,a+1)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$17$ |
17B |
$1$ |
\( 2^{2} \) |
$0.514595823$ |
$2.451929956$ |
3.582798921 |
\( -\frac{6357609}{131072} a + \frac{1888137}{4096} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -44 a + 648\) , \( -476 a + 7792\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a-1\right){x}^2+\left(-44a+648\right){x}-476a+7792$ |
| 32.2-b1 |
32.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{50} \) |
$2.39512$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2 \cdot 13 \) |
$0.496563266$ |
$1.153196483$ |
5.284566625 |
\( \frac{1121622319}{8192} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 336\) , \( 552 a - 108\bigr] \) |
${y}^2={x}^3+\left(a+1\right){x}^2+\left(a+336\right){x}+552a-108$ |
| 32.2-c1 |
32.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{28} \) |
$2.39512$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$6.140113858$ |
$5.007574023$ |
5.456723368 |
\( -\frac{29791}{4} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a\) , \( 0\bigr] \) |
${y}^2={x}^3+\left(-a-1\right){x}^2+a{x}$ |
| 32.2-c2 |
32.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{17} \) |
$2.39512$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.070056929$ |
$5.007574023$ |
5.456723368 |
\( \frac{2759}{16} a - \frac{26847}{16} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -5 a\) , \( 3 a + 20\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(a+1\right){x}^2-5a{x}+3a+20$ |
| 32.2-c3 |
32.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{17} \cdot 11^{12} \) |
$2.39512$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$3.070056929$ |
$5.007574023$ |
5.456723368 |
\( -\frac{2759}{16} a - \frac{3011}{2} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 35 a + 128\) , \( -19 a - 1708\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a+1\right){x}^2+\left(35a+128\right){x}-19a-1708$ |
| 32.2-c4 |
32.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
32.2 |
\( 2^{5} \) |
\( 2^{26} \) |
$2.39512$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$12.28022771$ |
$2.503787011$ |
5.456723368 |
\( \frac{133432831}{2} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 160\) , \( 96 a - 128\bigr] \) |
${y}^2={x}^3+\left(-a-1\right){x}^2+\left(a+160\right){x}+96a-128$ |
| 32.5-a1 |
32.5-a |
$2$ |
$17$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{30} \cdot 11^{12} \) |
$2.39512$ |
$(2,a), (2,a+1)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$17$ |
17B |
$1$ |
\( 2^{2} \) |
$0.514595823$ |
$2.451929956$ |
3.582798921 |
\( \frac{6357609}{131072} a + \frac{54062775}{131072} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 45 a + 572\) , \( 520 a + 7888\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(45a+572\right){x}+520a+7888$ |
| 32.5-a2 |
32.5-a |
$2$ |
$17$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{42} \) |
$2.39512$ |
$(2,a), (2,a+1)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$17$ |
17B |
$1$ |
\( 2^{2} \) |
$0.514595823$ |
$2.451929956$ |
3.582798921 |
\( -\frac{6357609}{131072} a + \frac{1888137}{4096} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -4\) , \( -2 a + 58\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2-4{x}-2a+58$ |
| 32.5-b1 |
32.5-b |
$4$ |
$4$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{28} \) |
$2.39512$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$6.140113858$ |
$5.007574023$ |
5.456723368 |
\( -\frac{29791}{4} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a\) , \( 0\bigr] \) |
${y}^2={x}^3+\left(a+1\right){x}^2+a{x}$ |
| 32.5-b2 |
32.5-b |
$4$ |
$4$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{17} \cdot 11^{12} \) |
$2.39512$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$3.070056929$ |
$5.007574023$ |
5.456723368 |
\( \frac{2759}{16} a - \frac{26847}{16} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -35 a + 163\) , \( 19 a - 1727\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-35a+163\right){x}+19a-1727$ |
| 32.5-b3 |
32.5-b |
$4$ |
$4$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{17} \) |
$2.39512$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.070056929$ |
$5.007574023$ |
5.456723368 |
\( -\frac{2759}{16} a - \frac{3011}{2} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -11 a + 9\) , \( 6 a + 77\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(-11a+9\right){x}+6a+77$ |
| 32.5-b4 |
32.5-b |
$4$ |
$4$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{26} \) |
$2.39512$ |
$(2,a), (2,a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$12.28022771$ |
$2.503787011$ |
5.456723368 |
\( \frac{133432831}{2} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 160\) , \( -96 a + 128\bigr] \) |
${y}^2={x}^3+\left(a+1\right){x}^2+\left(a+160\right){x}-96a+128$ |
| 32.5-c1 |
32.5-c |
$1$ |
$1$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
32.5 |
\( 2^{5} \) |
\( 2^{50} \) |
$2.39512$ |
$(2,a), (2,a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2 \cdot 13 \) |
$0.496563266$ |
$1.153196483$ |
5.284566625 |
\( \frac{1121622319}{8192} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 336\) , \( -552 a + 108\bigr] \) |
${y}^2={x}^3+\left(-a-1\right){x}^2+\left(a+336\right){x}-552a+108$ |
| 44.3-a1 |
44.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
44.3 |
\( 2^{2} \cdot 11 \) |
\( 2^{39} \cdot 11 \) |
$2.59360$ |
$(2,a), (2,a+1), (11,a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$1.511560784$ |
0.536517320 |
\( \frac{28986836667825}{184549376} a - \frac{11874309034897}{184549376} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -41 a - 56\) , \( 428 a - 624\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(-41a-56\right){x}+428a-624$ |
| 44.3-a2 |
44.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
44.3 |
\( 2^{2} \cdot 11 \) |
\( 2^{21} \cdot 11^{3} \) |
$2.59360$ |
$(2,a), (2,a+1), (11,a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$4.534682354$ |
0.536517320 |
\( \frac{15306417}{340736} a - \frac{21894673}{340736} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 4 a + 24\) , \( -4 a + 16\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(4a+24\right){x}-4a+16$ |
| 44.4-a1 |
44.4-a |
$2$ |
$3$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
44.4 |
\( 2^{2} \cdot 11 \) |
\( 2^{21} \cdot 11^{3} \) |
$2.59360$ |
$(2,a), (2,a+1), (11,a+7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$4.534682354$ |
0.536517320 |
\( -\frac{15306417}{340736} a - \frac{205883}{10648} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -6 a + 28\) , \( 3 a + 12\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-6a+28\right){x}+3a+12$ |
| 44.4-a2 |
44.4-a |
$2$ |
$3$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
44.4 |
\( 2^{2} \cdot 11 \) |
\( 2^{39} \cdot 11 \) |
$2.59360$ |
$(2,a), (2,a+1), (11,a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$1.511560784$ |
0.536517320 |
\( -\frac{28986836667825}{184549376} a + \frac{534766488529}{5767168} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 39 a - 97\) , \( -429 a - 196\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(39a-97\right){x}-429a-196$ |
| 47.1-a1 |
47.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
47.1 |
\( 47 \) |
\( 2^{12} \cdot 47 \) |
$2.63673$ |
$(47,a+12)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.594396067$ |
$9.945076868$ |
2.098177390 |
\( \frac{197}{47} a + \frac{86880}{47} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 4 a + 11\) , \( -5 a + 3\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(-a+1\right){x}^2+\left(4a+11\right){x}-5a+3$ |
| 47.2-a1 |
47.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
47.2 |
\( 47 \) |
\( 2^{12} \cdot 47 \) |
$2.63673$ |
$(47,a+34)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.594396067$ |
$9.945076868$ |
2.098177390 |
\( -\frac{197}{47} a + \frac{87077}{47} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -4 a + 15\) , \( 5 a - 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(-4a+15\right){x}+5a-2$ |
| 52.3-a1 |
52.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
52.3 |
\( 2^{2} \cdot 13 \) |
\( 2^{19} \cdot 11^{12} \cdot 13^{6} \) |
$2.70422$ |
$(2,a), (2,a+1), (13,a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3 \) |
$2.018693019$ |
$1.116323686$ |
4.799210535 |
\( \frac{305086149374671}{158164877312} a - \frac{2152282255725807}{158164877312} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( -347 a - 7087\) , \( -19130 a - 228606\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3+\left(-347a-7087\right){x}-19130a-228606$ |
| 52.3-a2 |
52.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
52.3 |
\( 2^{2} \cdot 13 \) |
\( 2^{44} \cdot 13^{3} \) |
$2.70422$ |
$(2,a), (2,a+1), (13,a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$4.037386039$ |
$1.116323686$ |
4.799210535 |
\( -\frac{757762455100869}{2359010787328} a + \frac{423975891898661}{2359010787328} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -a + 129\) , \( -77 a - 339\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-a+129\right){x}-77a-339$ |
| 52.3-a3 |
52.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
52.3 |
\( 2^{2} \cdot 13 \) |
\( 2^{17} \cdot 11^{12} \cdot 13^{2} \) |
$2.70422$ |
$(2,a), (2,a+1), (13,a+4)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3 \) |
$6.056079058$ |
$3.348971059$ |
4.799210535 |
\( \frac{16419025}{692224} a + \frac{38962771}{21632} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( 38 a + 448\) , \( 106 a - 1202\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3+\left(38a+448\right){x}+106a-1202$ |
| 52.3-a4 |
52.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
52.3 |
\( 2^{2} \cdot 13 \) |
\( 2^{28} \cdot 13 \) |
$2.70422$ |
$(2,a), (2,a+1), (13,a+4)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$12.11215811$ |
$3.348971059$ |
4.799210535 |
\( -\frac{120797325}{13312} a + \frac{1542318461}{13312} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -16 a - 16\) , \( 256\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+\left(a-1\right){x}^2+\left(-16a-16\right){x}+256$ |
| 52.4-a1 |
52.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
52.4 |
\( 2^{2} \cdot 13 \) |
\( 2^{19} \cdot 11^{12} \cdot 13^{6} \) |
$2.70422$ |
$(2,a), (2,a+1), (13,a+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3 \) |
$2.018693019$ |
$1.116323686$ |
4.799210535 |
\( -\frac{305086149374671}{158164877312} a - \frac{57724878323473}{4942652416} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( 346 a - 7433\) , \( 19129 a - 247735\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(346a-7433\right){x}+19129a-247735$ |
| 52.4-a2 |
52.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
52.4 |
\( 2^{2} \cdot 13 \) |
\( 2^{44} \cdot 13^{3} \) |
$2.70422$ |
$(2,a), (2,a+1), (13,a+8)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$4.037386039$ |
$1.116323686$ |
4.799210535 |
\( \frac{757762455100869}{2359010787328} a - \frac{10430830100069}{73719087104} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -14 a + 160\) , \( 222 a - 192\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+\left(-14a+160\right){x}+222a-192$ |
| 52.4-a3 |
52.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
52.4 |
\( 2^{2} \cdot 13 \) |
\( 2^{17} \cdot 11^{12} \cdot 13^{2} \) |
$2.70422$ |
$(2,a), (2,a+1), (13,a+8)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3 \) |
$6.056079058$ |
$3.348971059$ |
4.799210535 |
\( -\frac{16419025}{692224} a + \frac{1263227697}{692224} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -39 a + 487\) , \( -107 a - 1095\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(-39a+487\right){x}-107a-1095$ |
| 52.4-a4 |
52.4-a |
$4$ |
$6$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
52.4 |
\( 2^{2} \cdot 13 \) |
\( 2^{28} \cdot 13 \) |
$2.70422$ |
$(2,a), (2,a+1), (13,a+8)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$12.11215811$ |
$3.348971059$ |
4.799210535 |
\( \frac{120797325}{13312} a + \frac{88845071}{832} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( a\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(a+1\right){x}^2+a{x}$ |
| 68.3-a1 |
68.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
68.3 |
\( 2^{2} \cdot 17 \) |
\( 2^{17} \cdot 17^{2} \) |
$2.89180$ |
$(2,a), (2,a+1), (17,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$2.851794104$ |
6.073339349 |
\( -\frac{6335496481}{2312} a - \frac{60341676321}{2312} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 16 a\) , \( -92 a - 640\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(a+1\right){x}^2+16a{x}-92a-640$ |
| 68.3-a2 |
68.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
68.3 |
\( 2^{2} \cdot 17 \) |
\( 2^{22} \cdot 17 \) |
$2.89180$ |
$(2,a), (2,a+1), (17,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$5.703588209$ |
6.073339349 |
\( \frac{1042099}{1088} a - \frac{1268623}{1088} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -4 a\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^3+\left(a+1\right){x}^2-4a{x}$ |
| 68.4-a1 |
68.4-a |
$2$ |
$2$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
68.4 |
\( 2^{2} \cdot 17 \) |
\( 2^{17} \cdot 17^{2} \) |
$2.89180$ |
$(2,a), (2,a+1), (17,a+15)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$2.851794104$ |
6.073339349 |
\( \frac{6335496481}{2312} a - \frac{33338586401}{1156} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -32 a + 30\) , \( 122 a - 27\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(-32a+30\right){x}+122a-27$ |
| 68.4-a2 |
68.4-a |
$2$ |
$2$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
68.4 |
\( 2^{2} \cdot 17 \) |
\( 2^{22} \cdot 17 \) |
$2.89180$ |
$(2,a), (2,a+1), (17,a+15)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$5.703588209$ |
6.073339349 |
\( -\frac{1042099}{1088} a - \frac{56631}{272} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -12 a + 10\) , \( 10 a + 85\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}^2+\left(-12a+10\right){x}+10a+85$ |
| 88.3-a1 |
88.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
88.3 |
\( 2^{3} \cdot 11 \) |
\( 2^{15} \cdot 11^{15} \) |
$3.08433$ |
$(2,a), (2,a+1), (11,a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$25$ |
\( 3 \) |
$1$ |
$0.793522192$ |
1.173561804 |
\( \frac{2177104794557021}{2725888} a - \frac{32972319870461}{85184} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 13320 a + 14971\) , \( 493764 a + 7651060\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(13320a+14971\right){x}+493764a+7651060$ |
| 88.3-a2 |
88.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
88.3 |
\( 2^{3} \cdot 11 \) |
\( 2^{37} \cdot 11^{21} \) |
$3.08433$ |
$(2,a), (2,a+1), (11,a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$0.264507397$ |
1.173561804 |
\( -\frac{5506800748435829210747}{20254616437047427072} a + \frac{550956620077767845339}{632956763657732096} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 13865 a + 3771\) , \( 73570 a + 8857208\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+1\right){x}^2+\left(13865a+3771\right){x}+73570a+8857208$ |
| 88.6-a1 |
88.6-a |
$2$ |
$3$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
88.6 |
\( 2^{3} \cdot 11 \) |
\( 2^{37} \cdot 11^{21} \) |
$3.08433$ |
$(2,a), (2,a+1), (11,a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$0.264507397$ |
1.173561804 |
\( \frac{5506800748435829210747}{20254616437047427072} a + \frac{12123811094052741840101}{20254616437047427072} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -13867 a + 17636\) , \( -73571 a + 8930778\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-13867a+17636\right){x}-73571a+8930778$ |
| 88.6-a2 |
88.6-a |
$2$ |
$3$ |
\(\Q(\sqrt{-127}) \) |
$2$ |
$[0, 1]$ |
88.6 |
\( 2^{3} \cdot 11 \) |
\( 2^{15} \cdot 11^{15} \) |
$3.08433$ |
$(2,a), (2,a+1), (11,a+7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$25$ |
\( 3 \) |
$1$ |
$0.793522192$ |
1.173561804 |
\( -\frac{2177104794557021}{2725888} a + \frac{1121990558702269}{2725888} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -13322 a + 28291\) , \( -493765 a + 8144824\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-13322a+28291\right){x}-493765a+8144824$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
